I am a high school student and am trying to find the taylor expansion of $\tanh(x)$ in terms of a summation form. I have gotten this far, and am aware it might get complicated very quickly. If someone could aid in finishing it for me (or show a source (note I am still only a senior student so if you could not skip any steps).
I said:
$$\tanh(x) = c(1) + c(2)x + c(3)x^2 + c(4)x^3 + … c(n)x^{n-1}$$
$$\tanh(x)\cosh(x) = \sinh(x)$$
$$(c(1) + c(2)x + c(3)x^2 + c(4)x^3 + … )(1 + ((x^2)/2!) + ((x^4)/4!) + … ) = x + ((x^3)/3!) + ((x^5)/5!) + \ldots$$
then from there, I got that,
$$\tanh(x) = c(1) + c(2)(x) + (c(3)+(c(1)/(2!))(x^2) + (c(4)+(c(2)/(4!))(x^3) + (c(5)+(c(1)/(4!)+c(3)/(2!))(x^4) + (c(6)+(c(2)/(4!)+c(4)/(2!))(x^5) + (c(7)+(c(1)/(6!)+c(3)/(4!)+(c(5)/(2!))(x^6) + …$$
As is obvious there are a multitude of patterns, after each power of two progression of $x$, there is an extra way of attaining that power of $x$. (eg. in the first 2 terms of the expansion, there is only 1 way to attain that power of $x$, in the next 2 terms, there are 2 ways to attain that power of $x$, increasing by 1 every 2 terms.
I could name countless other obvious patterns though I don't think that is very useful.
I am thinking that pascal's triangle might come in the explanation somewhere however not sure where.
Also I am not sure what the Bernoulli series are which I think is important in order to find the summation expression, if someone could either find a summation expression which doesn't use Bernoulli's numbers or is willing to explain this to me, that would very generous of you.
Please keep it at high school level (and it doesn't matter if the final summation isn't simplified, as long as it is understandable), if there is something that needs to be introduced (such as the Bernoulli numbers), I will learn it (maybe provide a source which explains this to me in detail, would be nice).
Thank you so much, I will be extremely grateful, you have no idea how happy I will be once I finally get it, I just can't stop thinking about it!
Best Answer
Direct computation from the definition of Taylor series for $f(x)=\tanh x$ about $x=0$ gives $$ \tanh x=\color{blue}{x-{x^3\over 3}+{2x^5\over 15}-{17x^7\over 315}+\cdots}=\sum_{n=1}^\infty {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}\over (2n)!}, \quad |x|<{\pi\over 2}, $$ where $B_{2n}$ refers to the $2n$th Bernoulli number which can be defined in a variety of ways. I suggest reading about them here and choosing whichever definition is most accessible to you in order to generate them.
The the first expression in blue not sufficient, especially for a high school level? Or are you requiring a general formulation for the $n$th term as on the right above? Because if so, you will be forced to deal with the Bernoulli numbers.